Acceleration perpendicular to velocity in 2D

In summary, when the magnitude of acceleration is constant and it is perpendicular to velocity, speed remains constant. However, if the magnitude of acceleration is not constant, the speed will also vary. To prove this, one can use the work-kinetic energy theorem and the fact that if acceleration and velocity are perpendicular, the work done on the system is zero, which means kinetic energy (and speed) remains constant. This can also be deduced from the fact that in circular motion, acceleration and velocity are always perpendicular.
  • #1
tmpr
5
0
If the magnitude of acceleration is constant, and acceleration is perpendicular to velocity, is speed constant? Also, is speed not constant when the magnitude of acceleration is not constant? How would I show this?

I tried to do this:

If position is [tex]p(t)=(x(t),y(t))[/tex], then velocity is [tex]p'(t)=(x'(t),y'(t))[/tex] and acceleration is [tex]p''(t)=(x''(t),y''(t))[/tex]. If the magnitude of acceleration is constant, [tex]|p''(t)|=k[/tex]. If acceleration and velocity are perpendicular, [tex]p'(t) \cdot p''(t) = x'(t)x''(t) + y'(t)y''(t) = 0 [/tex].

But I'm stuck here.

How do I show [tex]|p'(t)|=c[/tex] for some constant?
 
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  • #2
If the acceleration is perpendicular that means your object is moving in a circular path. So there is a centripetal force and hence acceleration acting.

a=v2/r, so 'a' and 'r' are constants.
 
  • #3
rock.freak667 said:
If the acceleration is perpendicular that means your object is moving in a circular path. So there is a centripetal force and hence acceleration acting.

a=v2/r, so 'a' and 'r' are constants.

OK, but can you show me how you would deduce the fact that the object is moving with circular motion, given the assumption that acceleration is perpendicular to velocity?
 
  • #4
tmpr said:
OK, but can you show me how you would deduce the fact that the object is moving with circular motion, given the assumption that acceleration is perpendicular to velocity?

Force is in the direction of acceleration, meaning that normally the acceleration and velocity would lie in the same plane. As far as I know, the only time acceleration is perpendicular to velocity is during circular motion.
 
  • #5
Assuming that there is no external forces other than the one causing the acceleration.

Use the work-kinetic energy theorem:

In that case:

[tex]\Delta T =- \Delta W = -\int \vec{F}\cdot d\vec{r}[/tex]

From here you should be able to show that T doesn't change if a is always perpendicular to the velocity of the particle (write dr=v(t)dt), and thus speed stays the same.
 
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  • #6
If acceleration and velocity are perpendicular:
a1v1+a2v2=0
but that is nothing else than the time derivative of v2, that is speed is costant whenever a and v are perpendicular.

But I like G01 explanation. It focus on the physics of the system.
If force and velocity are perpendicular (forse perpendicular to the direction of motion), work made on the system is zero, which implies that the kinetic energy (and so speed) is costant.
 

1. What is acceleration perpendicular to velocity in 2D?

Acceleration perpendicular to velocity in 2D refers to the rate of change of the direction of an object's velocity in a two-dimensional space. It is the component of acceleration that is perpendicular to the direction of motion.

2. How is acceleration perpendicular to velocity in 2D calculated?

Acceleration perpendicular to velocity in 2D can be calculated using the formula a⊥ = (v²⊥ - v²∥)/r, where a⊥ is the perpendicular acceleration, v²⊥ is the perpendicular component of velocity, v²∥ is the parallel component of velocity, and r is the radius of curvature of the object's path.

3. What is the relationship between acceleration perpendicular to velocity and centripetal acceleration?

Acceleration perpendicular to velocity and centripetal acceleration are equivalent in magnitude and direction. They both represent the acceleration of an object towards the center of its circular motion.

4. How does acceleration perpendicular to velocity affect an object's motion?

Acceleration perpendicular to velocity causes an object to change its direction of motion, while maintaining a constant speed. This results in a curved path or circular motion for the object.

5. Can acceleration perpendicular to velocity ever be zero?

Yes, acceleration perpendicular to velocity can be zero if the object is moving in a straight line or if its velocity is constant. This means that there is no change in the direction of the object's motion.

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